Along with Carl, I was at least mildly intrigued by the parallel
with manumission rites. However, as Joseph Brian Tucker's's earlier
post noted, only the Gal 5:13 reference hasthe preposition found in the
manumission formula. It seems to me that we need to deal with 5:13
and 5:1 separately, due to the inclusion or absence of EPI.

Perhaps the opening two words of Gal 5:1 are a dative of manner,
rather than of purpose or result? "Christ has freely set us free."
This would fit well within the argument of the larger passage, as
follows:
"Christ set us free freely (i.e. our receipt of freedom was not
achieved through following a code or law); therefore, our free life
ought not be encumbered with any such law or code." Thus, the idea of
purpose is certainly present within the passage, but not within the
dative, itself.

As for the 5:13 occurrence, whether this truly fits the manumission
formula or not, the sense seems plain enough: Paul is here guarding the
"other side of the drunken man's horse." Having exhorted the Galatians
in 5:1 not to follow freedom with bondage, he then makes certain they
will not follow it with licentiousnss.

5:1, then, is a dative of manner describing the manner of the
freeing process; 5:13 has reference to the purpose or result of that
same manumittive act.